Computational Mathematics and Variational Analysis

Frontmatter

Scattering Relations of Elastic Waves by a Multi-Layered Thermoelastic Body

Abstract

The scattering problem of a time-harmonic dependent plane elastic wave by a multi-layered thermoelastic body in an isotropic and homogeneous elastic medium is considered. The direct scattering problem is formulated. Integral representations for the total exterior elastic field and the total interior thermoelastic fields as well as expressions for the far-field patterns are obtained containing the physical parameters of the interior thermoelastic layers. A reciprocity type theorem, a general type scattering theorem and an optical type theorem for plane wave incidence are presented and proved.

Evagelia S. Athanasiadou, Vassilios Sevroglou, Stefania Zoi

Blind Transfer of Personal Data Achieving Privacy

Abstract

Exploitation of data for statistical or economic analyses is an important and rapidly growing area. In this article, we address the problem of privacy when data containing sensitive information are processed by a third party. In order to solve this problem, we propose a cryptographic protocol and we prove its security. The security analysis leads to introduce the new notion of generalized discrete logarithm problem. Our protocol has effectively been deployed within a network of more than 5000 pharmacies.

Alexis Bonnecaze, Robert Rolland

Equilibria of Parametrized N-Player Nonlinear Games Using Inequalities and Nonsmooth Dynamics

Abstract

In this paper we present a combination of theoretical and computational results meant to give insights into the question of existence of non-unique Nash equilibria for N-players nonlinear games. Our inquiries make use of the theory of variational inequalities and projected systems to highlight cases where multiplayer Nash games with parametrized payoffs exhibit changes in the number of Nash equilibria, depending on given parameter values.

Monica G. Cojocaru, Fatima Etbaigha

Numerical Approximation of a Class of Time-Fractional Differential Equations

Abstract

We consider a class of linear fractional partial differential equations containing two time-fractional derivatives of orders α, β ∈ (0, 2) and elliptic operator on space variable. Three main types of such equations with α and β in the corresponding subintervals were determined. The existence of weak solutions of the corresponding initial-boundary value problems has been proved. Some finite difference schemes approximating these problems are proposed and their stability is proved. Estimates of their convergence rates, in special discrete energetic Sobolev’s norms, are obtained. The theoretical results are confirmed by numerical examples.

Aleksandra Delić, Boško S. Jovanović, Sandra Živanović

Approximating the Integral of Analytic Complex Functions on Paths from Convex Domains in Terms of Generalized Ostrowski and Trapezoid Type Rules

Abstract

In this paper we establish some results in approximating the integral of analytic complex functions on paths from convex domains in terms of generalized Ostrowski and Trapezoid type rules. Error bounds for these expansions in terms of p-norms are also provided. Examples for the complex logarithm and the complex exponential are also given.

Silvestru Sever Dragomir

Szász–Durrmeyer Operators and Approximation

Abstract

The Szász–Durrmeyer operators were introduced three and half decades ago in order to approximate integrable functions on the positive real axis. Several approximation properties of these operators have been discussed by researchers. In the present paper, we discuss some of the approximation properties of these operators in terms of weighted modulus of continuity and also in terms of first-order modulus of continuity having exponential growth. In the end, we find the difference estimate of Szász–Durrmeyer operators from the Baskakov–Szász–Mirakyan operators in weighted approximation.

Vijay Gupta

Leibniz’s Rule and Fubini’s Theorem Associated with a General Quantum Difference Operator

Abstract

In this paper, we derive Leibniz’s rule and Fubini’s theorem associated with a general quantum difference operator D _β which is defined by \(D_\beta f(t)=\frac {f(\beta (t))-f(t)}{\beta (t)-t}\), β(t) ≠ t. Here β is a strictly increasing continuous function defined on a set \(I\subseteq \mathbb R\) that has only one fixed point s ₀ ∈ I and satisfies the inequality (t − s ₀)(β(t) − t) ≤ 0 for all t ∈ I.

Alaa E. Hamza, Enas M. Shehata, Praveen Agarwal

Some New Ostrowski Type Integral Inequalities via General Fractional Integrals

Abstract

In this paper, authors discover an interesting identity regarding Ostrowski type integral inequalities. By using the lemma as an auxiliary result, some new estimates with respect to Ostrowski type integral inequalities via general fractional integrals are obtained. It is pointed out that some new special cases can be deduced from main results. Some applications to special means for different real numbers and new error estimates for the midpoint formula are provided as well. The ideas and techniques of this paper may stimulate further research.

Artion Kashuri, Themistocles M. Rassias

Some New Integral Inequalities via General Fractional Operators

Abstract

Trapezoidal inequalities for functions of diverse natures are useful in numerical computations. The authors have proved an identity for a generalized integral operator via differentiable function. By applying the established identity, the generalized trapezoidal type integral inequalities have been discovered. It is pointed out that the results of this research provide integral inequalities for almost all fractional integrals discovered in recent past decades. Various special cases have been identified. Some applications of presented results to special means have been analyzed and new error estimates for the trapezoidal formula are provided as well. The ideas and techniques of this paper may stimulate further research.

Artion Kashuri, Themistocles M. Rassias, Rozana Liko

Asymptotic Statistical Results: Theory and Practice

Abstract

The target of this paper is to discuss the existent difference of Asymptotic Theory in Statistics comparing to Mathematics. There is a need for a limiting distribution in Statistics, usually the Normal one. Adopting the sequential principle the first-order autoregression model and the stochastic approximation are referred for their particular interest for asymptotic results.

Christos P. Kitsos, Amílcar Oliveira

On the Computational Methods in Non-linear Design of Experiments

Abstract

In this paper the non-linear problem is discussed, for point and interval computational estimation. For the interval estimation an adjusted formulation is discussed due to Beale’s measure of non-linearity. The non-linear experimental design problem is regarded when the errors of observations are assumed i.i.d. and normally distributed as usually. The sequential approach is adopted. The average-per-observation information matrix is adopted to the developed theoretical approach. Different applications are discussed and we provide evidence that the sequential approach might be the panacea for solving a non-linear optimal experimental design problem.

Christos P. Kitsos, Amílcar Oliveira

Geometric Derivation and Analysis of Multi-Symplectic Numerical Schemes for Differential Equations

Abstract

In the current work we present a class of numerical techniques for the solution of multi-symplectic PDEs arising at various physical problems. We first consider the advantages of discrete variational principles and how to use them in order to create multi-symplectic integrators. We then consider the nonstandard finite difference framework from which these integrators derive. The latter is now expressed at the appropriate discrete jet bundle, using triangle and square discretization. The preservation of the discrete multi-symplectic structure by the numerical schemes is shown for several one- and two-dimensional test cases, like the linear wave equation and the nonlinear Klein–Gordon equation.

Odysseas Kosmas, Dimitrios Papadopoulos, Dimitrios Vlachos

Additive (ρ1, ρ2)-Functional Inequalities in Complex Banach Spaces

Abstract

In this paper, we introduce and solve the following additive (ρ ₁, ρ ₂)-functional inequalities:

$$\displaystyle \begin{aligned} \begin{array}{rcl} \left\|f\left(x-y\right) - f(x )+ f(y)\right\| &\displaystyle \ge &\displaystyle \|\rho_1 (f(x+y)-f(x)-f(y))\| \\ &\displaystyle + &\displaystyle \left\|\rho_2 \left( f(y-x)-f(y)+f(x)\right)\right\|, {} \end{array} \end{aligned} $$

(1)

where ρ ₁ and ρ ₂ are fixed complex numbers with |ρ ₁| + |ρ ₂| > 1, and

$$\displaystyle \begin{aligned} \begin{array}{rcl} \left\|f\left(x+y\right) - f(x )- f(y)\right\|&\displaystyle \ge &\displaystyle \|\rho_1 (f(x-y)-f(x)+f(y))\| \\ &\displaystyle + &\displaystyle \left\|\rho_2 \left( f(y-x)-f(y)+f(x)\right)\right\| ,{} \end{array} \end{aligned} $$

(2)

where ρ ₁ and ρ ₂ are fixed complex numbers with 1 + |ρ ₁| > |ρ ₂| > 1. Using the fixed point method and the direct method, we prove the Hyers–Ulam stability of the additive (ρ ₁, ρ ₂)-functional inequalities (2) and (1) in complex Banach spaces.

Jung Rye Lee, Choonkil Park, Themistocles M. Rassias

First Study for Ramp Secret Sharing Schemes Through Greatest Common Divisor of Polynomials

Abstract

A ramp secret sharing scheme through greatest common divisor of polynomials is presented. Verification and shelf correcting protocols are also developed. The proposed approach can be implemented in a hybrid way using numerical and symbolical arithmetic. Numerical examples illustrating the proposed sharing schemes are also given.

Gerasimos C. Meletiou, Dimitrios S. Triantafyllou, Michael N. Vrahatis

From Representation Theorems to Variational Inequalities

Abstract

We start with an historical introduction and then give an overview of the most important representation theorems for the linear(nonlinear) continuous functionals by the bifunction. Then we extensively study representations theorems for nonlinear operators. These representation results contain the difference of two(more) monotone operators, complementarity problems, systems of the absolute values equations and difference of two convex functions as special cases. These problems are very important and significant, which provide a unified and general framework of studying a wide class of unrelated problems.

Muhammad Aslam Noor, Khalida Inayat Noor

Unsupervised Stochastic Learning for User Profiles

Abstract

An unsupervised learning method for user profiles is examined. A user profile is considered the set of all the queries a user issues against an information or a database system. The mechanism of the Markovian model is employed where probabilistic locality translates to semantic locality in ways that facilitate a hierarchical clustering with optimal properties.

Nikolaos K. Papadakis

On the Solution of Boundary Value Problems for Ordinary Differential Equations of Order n and 2n with General Boundary Conditions

Abstract

We present a method for examining the existence and uniqueness and obtaining the exact solution to boundary value problems consisting of the differential equation Au = f, where A is a linear ordinary differential operator of order n, and multipoint and integral boundary conditions. We also derive a formula for computing the exact solution to even order boundary value problems encompassing the differential equation A ²u = f subject to 2n general boundary conditions. The method is based on the correct extensions of operators in Banach spaces.

I. N. Parasidis, E. Providas, S. Zaoutsos

Additive-Quadratic Functional Inequalities

Abstract

In this paper, we introduce and solve the following additive-quadratic s-functional inequalities:

$$\displaystyle \begin{aligned} \begin{array}{rcl} &\displaystyle &\displaystyle \left\|f\left(x+y\right) +f(x-y)-2f(x) - f(y) - f(-y)\right\| \\ &\displaystyle &\displaystyle \qquad \le \left\|s\left( 2 f\left(\frac{x{+}y}{2}\right){-}f(x){-}f(y){+} f\left(\frac{x{-}y}{2}\right) {+} f\left(\frac{y-x}{2}\right)\right)\right\| ,{} \end{array} \end{aligned} $$

(1)

where s is a fixed nonzero complex number with \(|s| <\sqrt {2}\), and

$$\displaystyle \begin{aligned} \begin{array}{rcl} &\displaystyle &\displaystyle \left\| 2 f\left(\frac{x+y}{2}\right)-f(x)-f(y)+ f\left(\frac{x-y}{2}\right) + f\left(\frac{y-x}{2}\right) \right\| \\ &\displaystyle &\displaystyle \qquad \le \left\|s\left(f\left(x+y\right) +f(x-y)-2f(x) - f(y) - f(-y)\right)\right\|,{} \end{array} \end{aligned} $$

(2)

where s is a fixed nonzero complex number with \(|s| <\frac {1}{2}\). Using the direct method and the fixed point method, we prove the Hyers–Ulam stability of the additive-quadratic s-functional inequalities (1) and (2) in complex Banach spaces.

Choonkil Park, Themistocles M. Rassias

Time-Delay Multi-Agent Systems for a Movable Cloud

Abstract

Adaptable splitting of calculations between movable devices and cloud is a significant and exciting research subject for movable cloud computing systems. Current work’s emphasis on the single-agent, neighbor-agent, or multi-agent systems, which aim to optimize the request achievement time for one particular agent. Due to the monopolistic competition for cloud resources among a large number of agents, the divested computations may be performed with a definite scheduling delay on the cloud. In this effort, we introduce three distributed dynamical systems, based on fractional calculus, describing the delay for single-agent, neighbor-agent, and multi-agent systems. We impose a new class of mixed-index time fractional differential equations. In addition, we investigate the asymptotic stability properties by using the fractional Cauchy’s method. We discuss the case of time-delay systems. A simulation is illustrated.

Rabha W. Ibrahim

The Global-Local Transformation

Abstract

The Global-Local transformation, a shape representation technique for manifolds of multiple dimensions is presented in this chapter. Useful properties of the transform space are examined and through experiments, unique advantages of the GL-transform are revealed. Applications in shape matching and radar shadow analysis demonstrate its effectiveness in real life scenarios.

Konstantinos A. Raftopoulos

On Algorithms for Difference of Monotone Operators

Abstract

This review proposes a proximal algorithm for difference of two monotone operators in finite dimensional real Hilbert space. Our route begins with reviewing some properties of DC (difference of convex functions) programming and DCA (DC algorithms). Next, we recall some main results about a proximal point algorithm for DC programming.

Maede Ramazannejad, Mohsen Alimohammady, Carlo Cattani

A Mathematical Model for Simulation of Intergranular μ-Capacitance as a Function of Neck Growth in Ceramic Sintering

Abstract

In this paper we will define a new mathematical model for predicting an evolution of an equivalent intergranular μ-capacitance during ceramic sintering. The contact between two adjacent grains will be defined as a structure that forms a μ-capacitor recognized as an intergranular μ-capacitor unit. It will be assumed that its μ-capacitance changes as the neck grows by diffusion. Diffusion mechanisms responsible for transport matter from the grain boundary to the neck are the volume diffusion and grain boundary diffusion. Such model does not need special geometric assumptions because the microstructural development can be simulated by a set of simple local rules and overall neck growth law which can be arbitrarily chosen. To find the total capacitance we will identify μ-capacitors in series and in parallel. More complicated connections of μ-capacitors will be transformed into simpler structure using delta to star transformation and/or star to delta transformation. In this way some μ-capacitors will be step-by-step replaced by their equivalent μ-capacitors. The developed model can be applied for the prediction of an evolution of the intergranular capacitance during ceramic sintering of BaTiO₃ system with spherical particle distributions.

Branislav M. Randjelović, Zoran S. Nikolić

Variational Inequalities in Semi-inner Product Spaces

Abstract

Variational Inequalities play an important role in solving many outstanding problems ranging from Mechanics, Physics, Engineering, and Economics. The work of the Italian and French mathematicians laid a solid mathematical foundation and today, it is an interesting area of considerable research activity. The variational inequalities were first considered in Hilbert spaces and subsequently to Banach spaces. In 1961, Lumer introduced the theory of semi-inner product spaces. This was followed by the work of Giles and many other researchers. In this paper, we have mentioned most of the results in variational inequalities in semi-inner product spaces. The new results proved in this paper are for a system of variational inequalities in a semi-inner product space. These results throw a light into the structural study of variational inequalities in uniformly smooth Banach spaces.

Nabin K. Sahu, Ouayl Chadli, R. N. Mohapatra

Results Concerning Certain Linear Positive Operators

Abstract

The present paper deals with some convergence results concerning Gupta-type operators. Varied sequences of linear positive operators have been discussed and their approximation properties been studied in literature. This work is a collection of these operators introduced over the past three decades.

Danyal Soybaş, Neha Malik

Behavior of the Solutions of Functional Equations

Abstract

In the last decades the oscillation theory of delay differential equations has been extensively developed. The oscillation theory of discrete analogues of delay differential equations has also attracted growing attention in the recent years. Consider the first-order delay differential equation,

$$\displaystyle \begin{aligned} x'(t) + p(t) \, x(\tau(t)) = 0, \,\,\,\,\,\, t \ge t_0, \end{aligned} $$

(1)

where \(p, \tau \in C([t_0, \infty ], \mathbb {R}^+)\), τ(t) is nondecreasing, τ(t) < t for t ≥ t ₀ and

$$\displaystyle \begin{aligned} \varDelta x(n) + p(t) \, x(\tau(n)) = 0, \,\,\,\,\,\, n = 0, 1, 2, \ldots , \end{aligned} $$

(2)

\( \mathop {\!\!\!\lim } \limits _{t \to \infty } \tau (t) =~\infty \), and the (discrete analogue) difference equation, where Δx(n) = x(n + 1) − x(n), p(n) is a sequence of nonnegative real numbers and τ(n) is a nondecreasing sequence of integers such that τ(n) ≤ n − 1 for all n ≥ 0 and \( \mathop {\lim } \limits _{n \to \infty } \tau (n) = \infty \). In this review chapter, a survey of the most interesting oscillation conditions is presented, along with numerical examples of delay and difference equations. We focus our attention on these examples, to illustrate the level of improvement in the oscillation criteria and the significance of the obtained results. The numerical calculations were made with the use of MATLAB software. These examples are relevant to many physical and biological applications.

Ioannis P. Stavroulakis, Michail A. Xenos

The Isometry Group of n-Dimensional Einstein Gyrogroup

Abstract

The space of n-dimensional relativistic velocities normalized to c = 1,

$$\displaystyle \mathbb {B} = \{\mathbf {v}\in \mathbb {R}^n\colon \|\mathbf {v}\| < 1\}, $$

is naturally associated with Einstein velocity addition ⊕_E, which induces the rapidity metric d _E on \(\mathbb {B}\) given by \(d_E(\mathbf {u}, \mathbf {v}) = \tanh ^{-1}\|-\mathbf {u}\oplus _E\mathbf {v}\|\). This metric is also known as the Cayley–Klein metric. We give a complete description of the isometry group of \((\mathbb {B}, d_E)\), along with its composition law.

Teerapong Suksumran

Function Variational Principles and Normed Minimizers

Abstract

The function variational principle due to El Amrouss [Rev. Col. Mat., 40 (2006), 1–14] may be obtained in a simplified manner. Further applications to existence of minimizers for Gâteaux differentiable bounded from below lsc functions over Hilbert spaces are then provided.

Mihai Turinici

Nadler-Liu Functional Contractions in Metric Spaces

Abstract

A technical extension is given for the fixed point result in Liu et al. [J. Appl. Math., Volume 2012, Article ID: 786061].

Mihai Turinici